CN113156365B - Speed, angle and distance joint estimation method based on conjugate ZC sequence pair - Google Patents

Speed, angle and distance joint estimation method based on conjugate ZC sequence pair Download PDF

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CN113156365B
CN113156365B CN202110272610.7A CN202110272610A CN113156365B CN 113156365 B CN113156365 B CN 113156365B CN 202110272610 A CN202110272610 A CN 202110272610A CN 113156365 B CN113156365 B CN 113156365B
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蒋轶
杨智宇
王锐
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Abstract

The invention belongs to the technical field of wireless positioning, and particularly relates to a speed, angle and distance joint estimation method based on a conjugate ZC sequence pair. The invention comprises the following steps: at a transmitting end, designing a pair of conjugate ZC sequences as a transmitting sequence; receiving multipath signals containing different transmission delays, Doppler frequency offsets and angles at a receiving end, and performing parameter estimation by using a maximum likelihood method; converting the high-dimensional parameter estimation problem into a plurality of low-dimensional parameter estimation problems by using an alternative projection method; for single-path parameter estimation, two-dimensional estimation of time delay and frequency offset is converted into two one-dimensional estimation based on the property of a ZC sequence, and then accurate estimation is carried out by combining Newton iteration. The invention can realize high-precision transmission delay, Doppler frequency offset and angle estimation under the condition of low bandwidth. Simulation shows that the invention can realize the distance estimation precision of 1cm, the speed estimation precision of 1m/s and the angle estimation precision of 0.01 DEG under the condition of 20MHz bandwidth.

Description

Speed, angle and distance joint estimation method based on conjugate ZC sequence pair
Technical Field
The invention belongs to the technical field of wireless positioning, and particularly relates to a speed, angle and distance joint estimation method based on a conjugate ZC sequence pair.
Background
The wireless ranging positioning algorithm is practically applied in life, and has wide application in the fields of indoor positioning, Internet of vehicles, automatic driving and the like. With the development of 5G and internet of things technologies, technologies related to positioning attract more and more attention. Market research companies, marks and marks, in the analytical reports published in 2020 predict that the market size for global indoor positioning will rise from $ 71.1 billion in 2017 to $ 409.9 billion in 2022 [1 ]. However, in a scene that some Global Positioning Systems (GPS) cannot cover or 5G automatic driving in the future, a technology capable of realizing high-resolution Positioning is urgently needed, and even centimeter-level Positioning accuracy is required.
Most of the existing research works only consider the joint estimation of time delay and angle, or cannot well distinguish multipath dense signals, or do not consider non-integral-multiple Nyquist sampling time delay. However, in practice, because the transceiving end has relative motion, doppler frequency offset exists, and better positioning can be performed by estimating the doppler frequency offset, namely, the velocity; at present, in the aspect of ranging accuracy, an Ultra Wide Band (UWB) technology can reach centimeter-level accuracy, but the technology needs a large bandwidth and has high requirements on hardware, so that under the condition of limited bandwidth, the non-integral-multiple nyquist sampling time delay estimation can improve the ranging accuracy and reduce the hardware cost; the time delay, Doppler frequency offset and angle of each path are considered, and the information is integrated to facilitate environment sensing, positioning and the like. Therefore, how to realize super-resolution estimation of time delay, Doppler frequency offset and angle in a multipath environment is an urgent problem to be solved.
Disclosure of Invention
The invention aims to provide a distance (transmission delay), angle and speed (Doppler frequency offset) joint estimation method in a wireless multipath environment with low operation complexity and low hardware implementation complexity.
The invention provides a distance, angle and speed joint estimation method under a wireless multipath environment.A pair of conjugate Zadoff-Chu sequences (ZC sequences) is designed at a transmitting end as a transmitting sequence; at a receiving end, receiving multipath signals containing different transmission delays, Doppler frequency offsets and angles, estimating parameters by using a maximum likelihood method, and then converting the original high-dimensional parameter estimation problem (multipath) into a plurality of low-dimensional parameter estimation problems (single path) by using an alternative projection method, thereby avoiding high-dimensional search; for single-path parameter estimation, two-dimensional estimation about time delay and frequency offset is converted into two one-dimensional estimation based on the property of a ZC sequence, the operation complexity is further reduced, and then accurate value estimation is carried out by combining Newton iteration.
The method comprises the following specific steps
Firstly, designing a pair of conjugate ZC sequences as a transmission sequence;
secondly, based on the property of the conjugate ZC sequence pair, simultaneously obtaining initial solutions of time delay, angle and Doppler frequency offset;
and thirdly, further obtaining an accurate value of the three-dimensional parameter through Newton iteration based on the initial solution of the time delay, the angle and the Doppler frequency offset.
Assuming that a receiving end has a uniform linear array with M antennas, the array response of the signal emitted to the array with respect to the angle θ can be expressed as:
Figure BDA0002975067120000021
where d represents the spacing of the antenna array, λ represents the wavelength, j represents the imaginary number, (. cndot.)TRepresenting a transpose operation.
Considering a multipath environment, a multipath signal received by a receiving end comprises 1 direct path and U-1 reflection paths, each path has different time delay, angle and Doppler frequency offset, and a multipath signal model is as follows:
Figure BDA0002975067120000022
wherein, betau，θu，τu，ξuRepresenting the channel gain, arrival angle, time delay and Doppler frequency offset of the u path signal; y (t) represents a received signal, x (t) represents a training sequence, z (t) represents Gaussian noise subject to a complex Gaussian distribution, U is the number of multipath signals,
Figure BDA00029750671200000216
representing a real number domain.
In the first step, a pair of conjugated Zadoff-Chu sequences is designed, and the specific process is as follows:
(1) for a length of
Figure BDA0002975067120000023
ZC sequence [2]：
Figure BDA0002975067120000024
Wherein,
Figure BDA0002975067120000025
is a positive integer, r is and
Figure BDA0002975067120000026
a co-prime positive integer parameter. It can be seen that
Figure BDA0002975067120000027
I.e., the ZC sequence is periodic, we can change the index range of the ZC sequence: for the
Figure BDA0002975067120000028
Even number, index range is changed to
Figure BDA0002975067120000029
To for
Figure BDA00029750671200000210
Is odd, the index range is
Figure BDA00029750671200000211
Suppose that
Figure BDA00029750671200000212
Is even, then for an integer delay τ there is:
Figure BDA00029750671200000213
this means that for a ZC sequence, an integer time delay τ corresponds
Figure BDA00029750671200000214
The frequency offset of (1); for the
Figure BDA00029750671200000215
The odd length, the time delay-frequency offset interconversion property of ZC sequences is still true. Without loss of generality, we let r be 1, so r can be omitted later.
(2) The time delay-frequency offset interchange property of the ZC sequence described in the formula (4) is only applicable to integer time delay, and in order to perform super-resolution time delay estimation, the influence of a shaping filter existing at a transmitting/receiving end needs to be considered. Assuming that the shaping filter existing at the transceiving end is a raised cosine filter, the impulse response of the raised cosine filter can be expressed as:
Figure BDA0002975067120000031
where α is the roll-off coefficient, TsIs the nyquist sampling period; the discrete ZC sequence s (n) after the shaping filter can be represented as a continuous time signal x (t):
Figure BDA0002975067120000032
due to the low-pass characteristic of the shaping filter, the high frequency part of the ZC sequence is suppressed. FIG. 1 is a
Figure BDA00029750671200000315
After the ZC sequence passes through a raised cosine filter, the middle low-frequency part is almost not influenced, and the high-frequency parts at two ends are obviously suppressed.
Further, it is demonstrated that the low frequency part with length L in the middle of the continuous-time signal x (t) is approximated as a chirp signal, that is:
Figure BDA0002975067120000033
wherein L is a positive integer and
Figure BDA0002975067120000034
it can also be seen from FIG. 1The roll-off coefficient α also affects the choice of L. Figure 2 shows that at a 0.3,
Figure BDA0002975067120000035
wherein | represents a modulo operation. It can be seen that for L-250, an approximation of equation (7) is appropriate.
Therefore, the signal of the low frequency part of the ZC sequence passing through the raised cosine filter can be regarded as a chirp signal, and there is also an interchange relationship between the time delay and the frequency offset, that is:
Figure BDA0002975067120000036
the delay τ may be arbitrary and is not limited to an integer multiple of the nyquist sampling period.
(3) Based on the approximation of equation (7), consider a pair of conjugated ZC sequences, denoted s (n) and s, respectively*(n)：
Figure BDA0002975067120000037
Figure BDA0002975067120000038
Wherein,
Figure BDA0002975067120000039
is an even number and is provided with a plurality of groups,
Figure BDA00029750671200000310
(·)*indicating a conjugate operation. For the first half of the ZC sequence s (n), add a length to it when transmitting
Figure BDA00029750671200000311
Prefix and length of
Figure BDA00029750671200000312
Suffix of (c):
Figure BDA00029750671200000313
Figure BDA00029750671200000314
wherein Q is a positive integer. Likewise, the second half of the ZC sequence is also prefixed and suffixed accordingly. The format of the transmitted training sequence is shown in fig. 3. The prefix and suffix are designed to resist the effects of intersymbol interference (ISI) and frequency offset. When the receiving end performs processing, part of the prefix and suffix are removed from the received signal. For example, in the first half of the training sequence, as shown in FIG. 4, the received L + Q long signal will be R long in time1ISI interference and frequency offset effects (equivalent length R)2Time offset) of the received signal, the receiving end performs prefix removal and postfix operations after receiving the signal, and selects the beginning of the truncated sequence from the region S, thereby ensuring that the truncated L-length sequence is not affected by ISI interference and frequency offset.
In the second step, the initial solution of the time delay, the angle and the doppler frequency offset is obtained, and the specific process is as follows:
(1) based on model equation (2), considering first half of the ZC sequence, the samples of the received signal can be expressed as:
Figure BDA0002975067120000041
wherein,
Figure BDA0002975067120000042
is an arbitrary real number greater than zero, TsIs the nyquist sampling period. Without loss of generality, let T s1, obtaining:
Figure BDA0002975067120000043
rewriting (14) to the form:
Y＝A(θ)diag(β)X(τ，ξ)T+Z (15)
wherein,
Figure BDA0002975067120000044
Figure BDA0002975067120000045
Figure BDA0002975067120000046
X(τ，ξ)＝[x(τ1)⊙d(ξ1)，x(τ2)⊙d(ξ2)，...，x(τU)⊙d(ξU)] (19)
Figure BDA0002975067120000047
Figure BDA0002975067120000048
wherein,
Figure BDA0002975067120000049
representing the complex field, diag (·) indicates that the vector is diagonalized into a matrix, and a-indicates a hadamard product.
Since the noise follows a complex gaussian distribution, the maximum likelihood estimate of the parameters β, τ, ξ, θ can be written in the form of least squares:
Figure BDA0002975067120000051
wherein τ ═ τ [ τ ]1，τ2，...，τU]T,ξ＝[ξ1，ξ2，...，ξU]T,θ＝[θ1，θ2，...，θU]T，||·||FRepresenting the F norm. Because of the fact that
Figure BDA0002975067120000052
Therefore, the method comprises the following steps:
Figure BDA0002975067120000053
where vec (-) represents matrix column vectorization,
Figure BDA0002975067120000054
which represents the kronecker product of,
Figure BDA0002975067120000055
Figure BDA0002975067120000056
therefore, (22) can be rewritten as:
Figure BDA0002975067120000057
wherein,
Figure BDA0002975067120000058
the maximum likelihood estimate of β can be expressed as:
Figure BDA0002975067120000059
wherein, (.)HIndicating a fetch transpose conjugate operation. Substituting (28) into (26) yields:
Figure BDA00029750671200000510
wherein,
Figure BDA00029750671200000511
wherein,
Figure BDA00029750671200000512
representing a projection matrix.
In estimating Doppler frequency offset
Figure BDA00029750671200000513
And time delay
Figure BDA00029750671200000514
After that, according to the speed
Figure BDA00029750671200000515
Where c is the speed of light, fcIs the carrier frequency, thereby estimating the speed of the transmitting end relative to the receiving end according to the distance
Figure BDA00029750671200000516
The distance can be estimated.
(2) Considering a conjugate ZC sequence pair, then:
Figure BDA00029750671200000517
wherein,
Figure BDA00029750671200000518
Figure BDA00029750671200000519
Figure BDA0002975067120000061
here, x1(τu，ξu) And
Figure BDA0002975067120000062
corresponding to the first half ZC sequence, x2(τu，ξu) And
Figure BDA0002975067120000063
corresponding to the second half ZC sequence.
Order:
Figure BDA0002975067120000064
Figure BDA0002975067120000065
and is
Figure BDA0002975067120000066
Is to be
Figure BDA0002975067120000067
From
Figure BDA0002975067120000068
The results obtained after deletion. Using the properties of the projection matrix, we obtain:
Figure BDA0002975067120000069
wherein,
Figure BDA00029750671200000610
representation matrix
Figure BDA00029750671200000611
The orthogonal projection matrix of (2).
Substituting equation (37) into equation (29) yields:
Figure BDA00029750671200000612
assume the multipath number U and
Figure BDA00029750671200000613
is known, i.e. given
Figure BDA00029750671200000614
Then equation (38) can be reduced to:
Figure BDA00029750671200000615
therefore, the multi-path problem is converted into a plurality of single-path problems by using an alternative projection method, and high-dimensional search is avoided.
(3) For the solution of the u-th path problem, firstly, the estimation of initial values of time delay, angle and frequency offset is carried out. The initial value of the parameter of the u-th path is achieved using the following approximation:
Figure BDA00029750671200000616
wherein,
Figure BDA00029750671200000617
due to the fact that
Figure BDA00029750671200000618
Then there are:
Figure BDA00029750671200000619
Figure BDA00029750671200000620
wherein,
Figure BDA0002975067120000071
and
Figure BDA0002975067120000072
are respectively
Figure BDA0002975067120000073
And
Figure BDA0002975067120000074
a matrix reconstructed column by column. Substituting (41) and (42) into (40) to obtain:
Figure BDA0002975067120000075
substituting (32) and (33) into (43) to obtain:
Figure BDA0002975067120000076
order:
Figure BDA0002975067120000077
then can be paired with τu，ξu，θuIs converted into pair etau，ζu，θuEstimation of (2):
Figure BDA0002975067120000078
considering the first half of the ZC sequence first, it is possible to obtain information about ζuAnd thetauSub-optimal solution of:
Figure BDA0002975067120000079
by pairs
Figure BDA00029750671200000710
Two-dimensional fast Fourier transform is carried out, and then the coordinate which enables the module value to be maximum is found out, thereby solving
Figure BDA00029750671200000711
Then the second half of the ZC sequence is considered, then
Figure BDA00029750671200000712
Known, there are:
Figure BDA00029750671200000713
at this time pair
Figure BDA00029750671200000714
One-dimensional fast Fourier transform is carried out to find out the coordinate which enables the module value to be maximum, thereby obtaining
Figure BDA00029750671200000715
Combining (47) (48) the estimation results to obtain
Figure BDA00029750671200000716
Further, initial values of the delay and the doppler frequency offset can be obtained:
Figure BDA00029750671200000717
in the third step, the accurate value of the three-dimensional parameter is further obtained through Newton iteration, and the specific process is as follows:
based on the obtained initial values
Figure BDA00029750671200000718
The invention utilizes Newton iteration [4]An accurate value is obtained. Let Λ denote the objective function (39), i.e.:
Figure BDA0002975067120000081
wherein ψ ═ τu，ξu，θu]T. The iteration of newton iterations is:
ψ(i+1)＝ψ(i)-sH-1g (51)
wherein,
Figure BDA0002975067120000082
and
Figure BDA0002975067120000083
the blackplug matrix and Jacobian vectors for the objective function Λ are represented, respectively, with s representing the step size, and by the back-tracking straight-line method [4 ]]And (6) optimizing to obtain.
The algorithm flow of the alternative projection method is as follows:
for the initialization of the alternating projection process, assuming that the received signal is single-path, in combination with equation (29), the following equation can be used to obtain the three-dimensional parameters:
Figure BDA0002975067120000084
wherein,
Figure BDA0002975067120000085
is constant and can therefore be omitted. Thereby estimating out
Figure BDA0002975067120000086
Wherein the superscript represents the number of iterations and thus is obtained
Figure BDA0002975067120000087
Order to
Figure BDA0002975067120000088
Estimated using equation (39)
Figure BDA0002975067120000089
Then, let
Figure BDA00029750671200000810
Thereby estimating
Figure BDA00029750671200000811
This process is continued until
Figure BDA00029750671200000812
Estimate out
Figure BDA00029750671200000813
At iteration 2, first use is made of
Figure BDA00029750671200000814
Estimated according to equation (39)
Figure BDA00029750671200000815
Then use
Figure BDA00029750671200000816
Estimate out
Figure BDA00029750671200000817
By analogy to utilize
Figure BDA00029750671200000818
Estimate out
Figure BDA00029750671200000819
The above process is repeated until the iterations converge.
The method has the advantages that:
(1) the invention can estimate the time delay, Doppler frequency offset and angle of each path from the multipath signal, thereby realizing distance measurement, speed measurement and direction measurement.
(2) The method considers the influence of the shaping filter in the real hardware and can carry out super-resolution time delay estimation.
(3) The invention avoids high-dimensional search through alternate iteration and has low complexity of hardware realization.
(4) The invention designs a transmitting sequence based on a ZC sequence, which can convert two-dimensional search about time delay and frequency offset into two one-dimensional search and reduce the complexity of operation.
(5) The method used in the present invention is superior to the existing SAGE 3 method in estimation accuracy.
The invention can realize high-precision transmission delay, Doppler frequency offset and angle estimation under the condition of low bandwidth, so as to estimate the motion speed of a transmitting end and the distance and angle between the transmitting end and a receiving end, and the information is integrated to be favorable for high-precision positioning. Simulation shows that the invention can realize the distance estimation precision of 1cm, the speed estimation precision of 1 m/and the angle estimation precision of 0.01 degrees under the condition of 20MHz bandwidth.
Drawings
FIG. 1 is a schematic diagram of signal angle of arrival with respect to a linear uniform array in a narrow-band far-field environment.
FIG. 2 is a plot of the modulus of a ZC sequence after passing through a shaping filter.
FIG. 3 is a graph showing that the approximation of equation (7) is good (for-125. ltoreq. t < 125, | x (t) -s (t) | < 1.8X 10-3).
Fig. 4 is a schematic diagram of a format of a transmitted training sequence based on a conjugate ZC sequence.
FIG. 5 is a schematic illustration of prefix and suffix removal.
FIG. 6 is a graph of RMSE performance for two-path velocity (a), angle (b), and distance (c) estimates.
Fig. 7 is a contour diagram of the direct path estimation performance for different delay and angle differences for the two paths (the first 3 (a), (b), (c)) and the corresponding cramer line (CRB) contour diagram (the last 3 (d), (e), (f)).
FIG. 8 is a graph of the performance of the method of the invention compared to the SAGE method (frequency offset).
FIG. 9 is a graph (angle) comparing the performance of the process of the present invention with that of SAGE.
FIG. 10 is a graph of the performance of the process of the present invention compared to that of SAGE (time delay).
Detailed Description
The invention is further illustrated by the following specific examples.
By way of example, the present invention emulates the complete process of signal transmit-shaping filter-pass channel-receive signal-signal processing with a computer. The conjugate ZC sequence pair L to be transmitted is 250,
Figure BDA0002975067120000095
each ZC sequence has a length
Figure BDA0002975067120000091
Prefix and length of
Figure BDA0002975067120000092
Suffix of (c). The shaping filter adopts a raised cosine filter, and the roll-off coefficient alpha is 0.3. If the receiving end has 6 antennae, the distance between the antennae is equal to half wavelength
Figure BDA0002975067120000093
Carrier frequency fc2.4GHz, bandwidth B20 MHz, i.e. T s50 ns. Each simulation was run 1000 times for monte carlo.
Example 1, considering a total of two paths, the frequency offset is ξ ═ 3 × 10-5，7×10-5]TsAngle θ is [5 °, 20 °)]Time delay is tau ═ 1.2, 1.3]Ts, channel gain
Figure BDA0002975067120000094
Wherein phi1And phi2Is a randomly generated random number. The transmission distance and speed of two paths are respectively corresponding to rho ═ 18, 19.5]m and v ═ 75, 175]m/s. FIG. 6 is a plot of the Root Mean Square Error (RMSE) curves of velocity, angle, and distance estimates of the two-path signals and their Clarmet plotsLead wire (CRB). Simulation shows that the method can effectively estimate the speed, the angle and the distance of the multipath signal, and can achieve the speed measurement precision of 1m/s, the distance measurement precision of 1cm and the angle measurement precision of 0.01 degrees.
In embodiment 2, a case where the estimation result changes with the change of the angle difference and the delay difference between the two paths is investigated, taking into account the two-path case with the same channel gain. Two-path frequency deviation xi ═ 10-5，10-5]/TsThe angle is [10 °, 10 ° + Δ θ ]]The time delay is [1.1, 1.1+ delta tau ═ 1.1]TsThe signal-to-noise ratio is 20 dB. The first 3 diagrams of fig. 7 show the RMSE results of frequency offset, angle, delay of the first path signal, and the last 3 diagrams show the corresponding CRB. The simulated contour lines show that the invention can distinguish multipath by using the difference in time and space even in the severe environment with dense multipath, and the simulation performance can approach to CRB.
Example 3, considering the two-path case of the same channel gain, the alternative projection method used in the present invention is compared with the SAGE method in terms of estimation accuracy. Two-path frequency deviation xi ═ 10-5，10-4]/TsAngle θ is [10 °, 15 ° ]]The time delay is [1.1, 1.1+ delta tau ═ 1.1]TsThe signal-to-noise ratio is 20 dB. Fig. 8, 9 and 10 show the RMSE results of the frequency offset, angle and time delay of the first path signal as the time delay difference between the two paths changes. Simulations show that the process of the invention is superior to SAGE: at 0.25TsFor example, SAGE method has frequency deviation, angle and time delay precision of 2 × 10-5/Ts(corresponding to a speed error of 50 m/s), 1.1 DEG, 0.06Ts(corresponding to a distance error of 90 cm); the method of the invention has the precision of frequency deviation, angle and time delay of 2.8 multiplied by 10-6/Ts(corresponding to a speed error of 7 m/s), 0.08 DEG, 4X 10-3Ts(corresponding to a distance error of 6 cm).
Reference to the literature
[1]Indoor Location Market by Component(Technology,Software Tools,and Services),Deployment Mode(Cloud,andOn-premises),Application,Vertical(Transportation,Hospitality,Entertainment,Retail,and Public Buildings),and Region-Global Forecastto2022[R],MarketAndMarket,2020.
[2]D.Chu,“Polyphase codes with good periodic correlation properties(corresp.),”IEEE Transactions on Information Theory,vol.18,no.4,pp.531–532,1972.
[3]B.H.Fleury,M.Tschudin,R.Heddergott,D.Dahlhaus,and K.Ingeman Pedersen,“Channel parameter estimation in mobile radio environments using the SAGE algorithm,”IEEE Journal on Selected Areasin Communications,vol.17,no.3,pp.434–450,1999.
[4]S.Boyd and L.Vandenberghe,Convex Optimization.2004。

Claims (3)

1. A speed, angle and distance joint estimation method based on conjugate ZC sequence pairs is characterized in that a pair of conjugate ZC sequences is designed at a transmitting end to be used as a transmitting sequence; at a receiving end, receiving multipath signals containing different transmission delays, Doppler frequency offsets and angles, estimating parameters by using a maximum likelihood method, and then converting the original high-dimensional parameter estimation problem into a plurality of low-dimensional parameter estimation problems by using an alternative projection method, thereby avoiding high-dimensional search; for single-path parameter estimation, two-dimensional estimation about time delay and frequency offset is converted into two one-dimensional estimation based on the property of a ZC sequence, so that the operation complexity is further reduced, and then accurate value estimation is carried out by combining Newton iteration; the method comprises the following specific steps:
firstly, designing a pair of conjugate ZC sequences;
secondly, based on the property of the conjugate ZC sequence pair, simultaneously obtaining initial solutions of time delay, angle and Doppler frequency offset;
thirdly, performing Newton iteration based on the initial solution of time delay, angle and frequency offset to further obtain the accurate values of three-dimensional parameters of time delay, angle and Doppler frequency offset;
wherein, assuming that a receiving end has a uniform linear array with M antennas, the array response of the signal to the array with respect to the angle θ is expressed as:
Figure FDA0003486565500000011
where d represents the spacing of the antenna array, λ represents the wavelength, j represents the imaginary number, (. cndot.)TRepresenting a transpose operation;
considering a multipath environment, a multipath signal received by a receiving end comprises 1 direct path and U-1 reflection paths, each path has different time delay, angle and Doppler frequency offset, and a multipath signal model is as follows:
Figure FDA0003486565500000012
wherein, betau,θu,τu,ξuRepresenting the channel gain, arrival angle, time delay and Doppler frequency offset of the u path signal; y (t) represents a received signal, x (t) represents a training sequence, and z (t) represents gaussian noise subject to a complex gaussian distribution; u is the number of multipath signals;
Figure FDA0003486565500000013
representing a real number domain;
in the second step, the initial solutions of the time delay, the angle and the Doppler frequency offset are obtained, and the specific process is as follows:
(1) based on model equation (2), considering the first half of the ZC sequence first, the samples of the received signal are expressed as:
Figure FDA0003486565500000014
wherein,
Figure FDA0003486565500000015
is an arbitrary real number greater than zero, TsIs the nyquist sampling period; let Ts1, obtaining:
Figure FDA0003486565500000016
rewriting (14) to the form:
Y＝A(θ)diag(β)X(τ,ξ)T+Z (15)
wherein,
Figure FDA0003486565500000021
Figure FDA0003486565500000022
Figure FDA0003486565500000023
X(τ,ξ)＝[x(τ1)⊙d(ξ1),x(τ2)⊙d(ξ2),…,x(τU)⊙d(ξU)] (19)
Figure FDA0003486565500000024
Figure FDA0003486565500000025
wherein,
Figure FDA0003486565500000026
representing a complex field, diag (·) represents vector diagonalization as a matrix operation, and an indicates a Hadamard product;
the noise follows a complex gaussian distribution, and the maximum likelihood estimate of the parameters β, τ, ξ, θ can be written in the form of least squares:
Figure FDA0003486565500000027
wherein τ ═ τ [ τ ]1,τ2,…,τU]T,ξ＝[ξ1,ξ2,…,ξU]T,θ＝[θ1,θ2,…,θU]T,‖·‖FRepresents the F norm;
because of the fact that
Figure FDA0003486565500000028
Where vec (-) represents matrix column vectorization,
Figure FDA0003486565500000029
representing the kronecker product, so there are:
Figure FDA00034865655000000210
wherein,
Figure FDA00034865655000000211
Figure FDA00034865655000000212
therefore, (22) is rewritten as:
Figure FDA00034865655000000213
wherein |2Which represents the 2-norm of the vector,
Figure FDA00034865655000000214
the maximum likelihood estimate of β is then expressed as:
Figure FDA0003486565500000031
wherein, (.)HRepresenting a transposed conjugation operation, substituting (28) into (26) yields:
Figure FDA0003486565500000032
wherein,
Figure FDA0003486565500000033
after estimating the Doppler frequency offset and time delay, according to the velocity
Figure FDA0003486565500000034
Where c is the speed of light, fcIs the carrier frequency, thereby estimating the speed of the transmitting end relative to the receiving end according to the distance
Figure FDA0003486565500000035
The distance can be estimated;
(2) considering a conjugate ZC sequence pair, then:
Figure FDA0003486565500000036
wherein,
Figure FDA0003486565500000037
to represent
Figure FDA0003486565500000038
The projection matrix of (2);
Figure FDA0003486565500000039
Figure FDA00034865655000000310
Figure FDA00034865655000000311
here, x1(τu,∫u) And
Figure FDA00034865655000000312
corresponding to the first half ZC sequence, x2(τu,ξu) And
Figure FDA00034865655000000313
corresponding to the second half ZC sequence;
order:
Figure FDA00034865655000000314
Figure FDA00034865655000000315
and is
Figure FDA00034865655000000316
Is to be
Figure FDA00034865655000000317
From
Figure FDA00034865655000000318
Deleting the result obtained; using the properties of the projection matrix, we obtain:
Figure FDA00034865655000000319
wherein,
Figure FDA00034865655000000320
representation matrix
Figure FDA00034865655000000321
The orthogonal projection matrix of (a);
substituting equation (37) into equation (29) yields:
Figure FDA00034865655000000322
assume the multipath number U and
Figure FDA00034865655000000323
is known, i.e. given
Figure FDA00034865655000000324
Then equation (38) is reduced to:
Figure FDA0003486565500000041
wherein, | · | represents modulo; therefore, the multi-path problem is converted into a plurality of single-path problems by using an alternative projection method;
(3) for the solution of the u-th path problem, firstly, estimating initial values of time delay, angle and frequency offset; the initial value of the parameter for the u-th path is achieved using the following approximation:
Figure FDA0003486565500000042
wherein,
Figure FDA0003486565500000043
due to the fact that
Figure FDA0003486565500000044
Then there are:
Figure FDA0003486565500000045
Figure FDA0003486565500000046
wherein,
Figure FDA0003486565500000047
and
Figure FDA0003486565500000048
are respectively
Figure FDA0003486565500000049
And
Figure FDA00034865655000000410
a matrix reconstructed by columns; substituting (41) and (42) into (40) to obtain:
Figure FDA00034865655000000411
substituting (32) and (33) into (43) to obtain:
Figure FDA00034865655000000412
order:
Figure FDA00034865655000000413
then, will be for τu,ξu,θuIs converted into pair etau,ζu,θuEstimation of (2):
Figure FDA00034865655000000414
first, considering the first half ZC sequence, obtain information about zetauAnd thetauSub-optimal solution of:
Figure FDA0003486565500000051
by pairs
Figure FDA0003486565500000052
Two-dimensional fast Fourier transform is carried out, and then the coordinate which enables the module value to be maximum is found out, thereby solving
Figure FDA0003486565500000053
Consider the second half of the ZC sequence again, at this point
Figure FDA0003486565500000054
Known, there are:
Figure FDA0003486565500000055
at this time pair
Figure FDA0003486565500000056
One-dimensional fast Fourier transform is carried out to find out the coordinate which enables the module value to be maximum, thereby obtaining
Figure FDA0003486565500000057
Combining (47) and (48) the estimation results to obtain
Figure FDA0003486565500000058
Further obtaining initial values of time delay and Doppler frequency offset:
Figure FDA0003486565500000059
。
2. the joint estimation method according to claim 1, wherein the designing of a pair of conjugate ZC sequences in the first step is performed as follows:
(1) for a length of
Figure FDA00034865655000000510
The ZC sequence of (1):
Figure FDA00034865655000000511
wherein,
Figure FDA00034865655000000512
is a positive integer, r is and
Figure FDA00034865655000000513
a co-prime positive integer parameter;
Figure FDA00034865655000000514
i.e., the ZC sequence is periodic, the index range of the ZC sequence can be changed: suppose that
Figure FDA00034865655000000515
Is even, then for an integer delay τ there is:
Figure FDA00034865655000000516
this means that for a ZC sequence, an integer time delay τ corresponds
Figure FDA00034865655000000517
The frequency offset of (1); for the
Figure FDA00034865655000000518
The length is odd, and the time delay-frequency offset interconversion property of the ZC sequence still holds; without loss of generality, let r equal to 1, i.e. r can be omitted;
(2) considering the influence of the shaping filter existing at the transmitting and receiving ends, assuming that the shaping filter existing at the transmitting and receiving ends is a raised cosine filter, the impulse response of the raised cosine filter is expressed as:
Figure FDA00034865655000000519
where α is the roll-off coefficient, TsIs the nyquist sampling period; the discrete ZC sequence s (n) is represented as a continuous time signal x (t) after passing through a shaping filter:
Figure FDA0003486565500000061
due to the low-pass characteristic of the shaping filter, the high-frequency part of the ZC sequence is suppressed; the low frequency part of length L in the middle of the continuous-time signal x (t) is approximated as a chirp signal, i.e.:
Figure FDA0003486565500000062
wherein L is a positive integer and
Figure FDA0003486565500000063
the roll-off coefficient α also affects the selection of L;
therefore, the signal of the low frequency part of the ZC sequence passing through the raised cosine filter is regarded as a chirp signal, and there is also an interchange relationship between the time delay and the frequency offset, that is:
Figure FDA0003486565500000064
the time delay tau can be any and is not limited to be an integral multiple of Nyquist sampling period;
(3) based on the approximation of equation (7), consider a pair of conjugated ZC sequences, denoted s (n) and s, respectively*(n)：
Figure FDA0003486565500000065
Figure FDA0003486565500000066
Wherein,
Figure FDA0003486565500000067
is an even number and is provided with a plurality of groups,
Figure FDA0003486565500000068
(·)*representing a conjugate taking operation; for the first half ZC sequence s (n), a prefix and a suffix with the total length of Q are added when a transmitting end transmits the sequence, and the prefix length is
Figure FDA0003486565500000069
And the suffix length is
Figure FDA00034865655000000610
Figure FDA00034865655000000611
Figure FDA00034865655000000612
Wherein Q is a positive integer; similarly, the second half of the ZC sequence is also added with a corresponding prefix and suffix; when the receiving end processes, part of the prefix and suffix are removed from the received signal.
3. The joint estimation method according to claim 1, characterized in that in the third step, the accurate values of the three-dimensional parameters are further obtained through newton iteration, as follows:
based on the obtained initial values
Figure FDA00034865655000000613
Obtaining an accurate value by utilizing Newton iteration; let Λ denote the objective function (39), i.e.:
Figure FDA00034865655000000614
wherein ψ ═ τu,ξu,θu]T(ii) a The iteration of newton iterations is:
ψ(i+1)＝ψ(i)-sH-1g (51)
wherein,
Figure FDA0003486565500000071
and
Figure FDA0003486565500000072
respectively representing a blackplug matrix and a Jacobian vector related to a target function Lambda, and s represents a step length and is obtained by a backtracking straight line method optimization;
the algorithm flow of the alternative projection method is as follows:
when the alternate projection is initialized, assuming that the received signal is single-path, in combination with equation (29), the following equation is used to obtain the three-dimensional parameters:
Figure FDA0003486565500000073
wherein,
Figure FDA0003486565500000074
is constant, omitted; thereby estimating out
Figure FDA0003486565500000075
Wherein the superscript (. circle.) represents the number of iterations, thus, obtaining
Figure FDA0003486565500000076
Order to
Figure FDA0003486565500000077
Estimated using equation (39)
Figure FDA0003486565500000078
Then, let
Figure FDA0003486565500000079
Thereby estimating
Figure FDA00034865655000000710
This process is continued until
Figure FDA00034865655000000711
Estimate out
Figure FDA00034865655000000712
At iteration 2, first use is made of
Figure FDA00034865655000000713
Estimated according to equation (39)
Figure FDA00034865655000000714
Then use
Figure FDA00034865655000000715
Estimate out
Figure FDA00034865655000000716
By analogy to utilize
Figure FDA00034865655000000717
Estimate out
Figure FDA00034865655000000718
The above process is repeated until the iterations converge.
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